nLab horizontal categorification

Idea

Horizontal categorification or Oidification describes the process by which

1. a concept is realized to be equivalent to a certain type of category or magmoid with a single object;

2. and then this concept is generalized – or oidified – by passing to instances of such types of categories with more than one object.

Remarks

• This is to be contrasted with vertical categorification.

• It can be argued that the term ‘categorification’ should be reserved for vertical categorification, since we can use ‘oidification’ for the horizontal concept.

• It has rightly been remarked that groupoids are more fundamental than groups, algebroids are more fundamental than algebras, etc. Hence in a better world, the suffix would be characterizing the one-object special cases, not the general concepts.

Examples

• The horizontal categorification of groups are groupoids: categories in which every morphism is invertible.

• A horizontal categorification of algebras are algebroids: enriched categories in the category of vector spaces.

• A horizontal categorification of rings are ringoids: enriched categories over the category of abelian groups. (blog)

• A horizontal categorification of $C^*$-algebras hence ought to be known as $C^*$–algebroids but is usually known as C*-categories.

• Since, by the Gelfand-Naimark theorem, C-star algebras are dual to topological spaces, Paolo Bertozzini et. al proposed to define spaceoids to be entities dual to $C^*$-categories (blog).

• And an exception to the rule: a many-object monoid is not called a monoidoid – but is called a category!

  On the other hand, if the category is $R$-linear (enriched in R Mod) then it may again be referred to as an algebroid over $R$.

• A horizontal categorification of the notion of monads is that of categories enriched in a bicategory.

• A horizontal categorification of a single-sorted Lawvere theory is a multisorted Lawvere theory. Analogously an operad categorifies to a many-colored operad.

• In type theory/programming language theory, horizontal categorification is analogous to introducing a type distinction: an untyped language is a special case of a typed language where there is exactly one type. This fits with the categorical semantics: untyped lambda calculus has models in Lawvere theories, whereas (simply) typed lambda calculus has models in Cartesian multicategories.

algebraic structure	oidification
truth value	preorder
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory

Further discussion

Related $n$-Café-discussion is in

• John Baez, What is categorification?

• John Baez, Ringoids